Given a finite sequence U_N={u_1,…,u_N} of points contained in the d-dimensional unit torus, we consider the L^2 discrepancy between the integral of a given function and the Riemann sums with respect to translations of U_N. We show that with positive probability, the L^2 discrepancy of other sequences close to U_N in a certain sense preserves the order of decay of the discrepancy of U_N. We also study the role of the regularity of the given function.
Brandolini, L., Chen, W., Gigante, G., Travaglini, G. (2009). Discrepancy for randomized riemann sums. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 137, 3177-3185.
Discrepancy for randomized riemann sums
TRAVAGLINI, GIANCARLO
2009
Abstract
Given a finite sequence U_N={u_1,…,u_N} of points contained in the d-dimensional unit torus, we consider the L^2 discrepancy between the integral of a given function and the Riemann sums with respect to translations of U_N. We show that with positive probability, the L^2 discrepancy of other sequences close to U_N in a certain sense preserves the order of decay of the discrepancy of U_N. We also study the role of the regularity of the given function.
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